Abstract
The phase spaces of 2-dim Mechanics and of 3-dim Optics are locally the same, but globally different. While momentum is an unbounded variable in classical mechanics, in geometric optics it is constrained to a disk that is the projection of the ray direction sphere, whose radius is the local index of refraction. Canonical transformations in mechanics preserve the Heisenberg-Weyl algebra. In geometric optics, in addition, they must preserve the natural momentum range. We show that the physical phenomena of optics that produce canonical transformations are actually much richer than those in mechanics: they include refracting surfaces. In fact, through the opening coma map, the range limitation may be lifted. The quadratic Hamiltonians of mechanics provide the paraxial régime of optics. The optical aberrations of the metaxial régime are the higher-order approximations. Finally, we note that 2-dim mechanics is based on the Heisenberg-Weyl group W2 while 3-dim optics is based on the Euclidean group ISO(3).
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Wolf, K.B. (1992). Canonical Transformations in Mechanics vis-à-vis Those in Optics. In: Frank, A., Wolf, K.B. (eds) Symmetries in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77284-9_22
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DOI: https://doi.org/10.1007/978-3-642-77284-9_22
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